Comments

Shouting "moron" is easy, throwing around technical terms, too. Getting it right? Not so easy.

<geek>

The problem is a bit deeper than rage is letting on. In order to define the length of a curve you need to have a (piecewise [1]) smooth [2] curve to begin with.

Is our target curve, the circle, smooth? Check.

Is each of the polygons that is used to approximate the circle piecewise smooth? Check.

Do the polygons converge to the circle? Check. (Despite rage shouting "moron", the "dropout" is correct in this)

What's the perimeter of the circle? Pi.

What's the perimeter of the polygons? 4.

Ooops, what's gone wrong? Look at the direction of the circle tangent and the direction of the tangents to the polygons. The tangent to the circle varies continuously between vertical and horizontal, whereas the tangents of the polygons in any one point are either vertical or horizontal. There are exactly four points on the circle, where the tangents of the polygon series converges to the tangent of the circle, where its tangents converge at all; those are the points at the top, left, right and bottom of the circle, where the tangents happen to be horizontal or vertical. In fact, those are the only points, where the tangents of the polygons converge, at all.

In short, the perimeter of the polygon series does not converge to the perimeter of the circle because its tangents do not converge to the tangents of the circle. You'd never know, if you just relied on rage.

</geek>

[1] i.e. for every area with a finite diameter that covers part of the curve the curve can devided into a finite number of pieces that each have the desired characteristic.[2] smooth as in smooth manifold, but here differentiable will do.

That comment might as well have been written in Tierra del Fuegan, which is the reason I'm a linguist and not a mathematician. Still, sorting through all that business about converging tangents and smooth manifolds, I still think the simplest explanation is best, at least for a Wolf of Very Little Brain.

A curve, by definition, has no angles, even at infinite magnification. The polygons can never truly equal the curve, but only approximate it - and the sum of the sides of the polygons will always equal four.

The original rage comic ended with the Archimedes panel. I added the rest, so I'm the one shouting "moron."

But every time I drive over a bridge, I'm grateful for people who can understand all the squiggles.

Ouch, wasn't aware you were the author of the "moron" shouting part. Sorry, I have misplaced my Feuerlandwörterbuch; I'll try once more in English.

You basically have the right idea, the polygons used do not approximate the circle in a way, that allows to determine the length of the circle circumference, but it's for a different reason than you thought, I think.

Let's start afresh: What is required for a curve to have a measurable length? Basically, you need to have a tangent at every point of the curve, i.e. for any given point on the curve there must exist a small neighborhood of that point, where the curve looks almost like a straight line. At latest at some microscopic level the curve must no longer twist and turn, however much you have to "magnify".

In order to find the length of a curve by consecutive approximation, your approximations must not only converge to the location of the curve, but its tangents must converge to the tangents of the curve in every point. Going back to the comic, to approximate the circle you can take a series of curves that gets closer and closer to the circle. The polygons used in the comic do a good job of approximating the place of the circle circumference. However, they do a lousy job at approximating the tangents of the circle, and that's why they fail to approximate the length of the circle.

Repeating steps an infinite number of times [1], as implied in the first few panels, can lead to unintuitive results. Every single member of the series may have some specific property, yet the limit structure does not need to have the same property. In this case, each of the approximating polygons has length 4, whereas the circle has a circumference of pi.

What if we wanted to do it properly? Just approximate the circle by arc segments, where the arc segments get smaller and smaller. This type of polygon would not only converge to the location of the circle, but its tangents would converge to the tangents of the circle.

Mein Lieber, as soon as you used the word "Tangent" you lost me. I did OK with Analytical Geometry, but it was all "by the book" - in other words, I never really "got" it. Yes, I know what a tangent is, and I can Google trig as well as anyone else, but there's something about the whole ball o' wax that forms an almost palpable barrier to understanding. I've given it up for a lost cause. I could learn Sanskrit faster than I could really understand higher math... and it would be more fun.

I'll try with the tangent bit. Its the straight line touching the circle as if it was the circle at the point where it touches (indeed it IS at the small point where it does touch) and extending out in either direction. Therefore if you draw an infinite number of those lines you'll end up with the circle. The lines making the angles as you make more and more corners NEVER go in the same direction as the circle where they touch so they don't count.